On the Number of Functions Realized by Cascades and Disjunctive Networks

In this paper, the number of functions realized by certain networks of two-input one-output gates are presented. Two networks are considered; one is the disjunctive network, which is characterized by the restriction that each gate output and each network input connect to exactly one gate input. The other network, the cascade, is the special case of the disjunctive networks in which each gate has at least one input which connects to a network input. For both networks, a recursion relation is derived for the number of realized switching functions dependent on exactly k variables. Both expressions have been solved by computer for k up to 15. Also, expressions are derived for the number of functions realized by cascades and disjunctive networks of two-input one-output cells, where each cell realizes any of the 16 functions on two variables.